how do you start this problem:
integral of xe^(-2x)
There are two ways:
1) Integration by parts.
2) Differentiation w.r.t. a suitably chosen parameter.
Lets do 1) first. This is the "standard method", but it is often more tedious than 2)
You first write the integral as:
Inegral xe^(-2x) dx =
Integral -1/2 x d(e^(-2x))
Here we have used that:
d(e^(-2x)) = -2 e^(-2x)
The next is is to make use of the fact that:
d(f g) = f dg + g df --->
f dg = d(fg) - g df
This yields:
Integral -1/2 x d(e^(-2x)) =
Integral d[-1/2 x e^(-2x)] -
Integral -1/2 e^(-2x) dx =
-1/2 x e^(-2x) - 1/4 e^(-2x) + C
Method 2) is much simpler. Consider the function:
e^(ax)
It's integral is:
Integral e^(ax)dx = 1/a e^(ax)
Le's differentiate both sides w.r.t. a:
Integral x e^(ax)dx =
[ -1/a^2 + x/a] e^(ax)
And insert a = -2 to obtain the answer.
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